The Kakimizu complex MS(K) for a knot K ⊂ S 3 is the simplicial complex with vertices the isotopy classes of minimal genus Seifert surfaces in the exterior of K and simplices any set of vertices with mutually disjoint representative surfaces. In this paper we determine the structure of the Kakimizu complex MS(K) of genus one hyperbolic knots K ⊂ S 3 . In contrast with the case of hyperbolic knots of higher genus, it is known that the dimension d of MS(K) is universally bounded by 4, and we show that MS(K) consists of a single d-simplex for d = 0, 4 and otherwise of at most two d-simplices which intersect in a common (d − 1)face. For the cases 1 ≤ d ≤ 3 we also construct infinitely many examples of such knots where MS(K) consists of two d-simplices.
We consider the question of how many essential Seifert Klein bottles with common boundary slope a knot in S 3 can bound, up to ambient isotopy. We prove that any hyperbolic knot in S 3 bounds at most six Seifert Klein bottles with a given boundary slope. The Seifert Klein bottles in a minimal projection of hyperbolic pretzel knots of length 3 are shown to be unique and π 1 -injective, with surgery along their boundary slope producing irreducible toroidal manifolds. The cable knots which bound essential Seifert Klein bottles are classified; their Seifert Klein bottles are shown to be non-π 1 -injective, and unique in the case of torus knots. For satellite knots we show that, in general, there is no upper bound for the number of distinct Seifert Klein bottles a knot can bound.
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